From d4e16e4b0eb59d37ed7bc6171af8aa230c6dc22d Mon Sep 17 00:00:00 2001
From: Emmanuel Fromager <manu@dhcp-35-195.u-strasbg.fr>
Date: Wed, 12 Feb 2020 12:04:03 +0100
Subject: [PATCH] Manu: saving work

---
 Manuscript/eDFT.tex | 67 +++++++++++++++++++++++++++++----------------
 1 file changed, 44 insertions(+), 23 deletions(-)

diff --git a/Manuscript/eDFT.tex b/Manuscript/eDFT.tex
index 6b62464..7a2516b 100644
--- a/Manuscript/eDFT.tex
+++ b/Manuscript/eDFT.tex
@@ -379,8 +379,8 @@ c}\left[n\right]}{\partial w_K}\right|_{n=n_{\bmg^{\bw}}}
 ,
 \eeq
 where ${\bm
-h}\equiv\langle\AO{\mu}\vert-\frac{1}{2}\nabla_{\br}^2+v_{\rm
-ne}(\br)\vert\AO{\nu}\rangle$ and ${\bm G}\equiv{\bm J}-{\bm K}$ denote
+h}\equiv\left\{\langle\AO{\mu}\vert-\frac{1}{2}\nabla_{\br}^2+v_{\rm
+ne}(\br)\vert\AO{\nu}\rangle\right\}_{\mu\nu}$ and ${\bm G}\equiv{\bm J}-{\bm K}$ denote
 the Coulomb-exchange
 integrals. 
 %%%%%%%%%%%%%%%%%%%%%
@@ -419,6 +419,46 @@ w}_K
 
 \subsection{Approximations}
 
+As Hartree and exchange energies cannot be separated in the
+one-dimension systems considered in the rest of this work, we will substitute the Hartree--Fock
+density-matrix-functional interaction energy,
+\beq\label{eq:eHF-dens_mat_func}
+W_{\rm
+HF}\left[{\bmg}\right]=\frac{1}{2} \Tr(\bmg \, \bG \, \bmg), 
+\eeq
+for the Hx density-functional energy in the variational energy
+expression of Eq.~(\ref{eq:var_ener_gokdft}):  
+\beq
+{\bmg}^{\bw}\approx\argmin_{{\bm\gamma}^{\bw}}
+\Big\{
+{\rm
+Tr}\left[{\bm \gamma}^{{\bw}}{\bm h}\right]+W_{\rm
+HF}\left[{\bm\gamma}^{\bw}\right]
++
+{E}^{{\bw}}_{\rm
+c}\left[n_{\bm\gamma^{\bw}}\right]
+%+E^{{\bw}}_{\rm c}\left[n_{\hat{\Gamma}^{{\bw}}}\right]
+\Big\}.
+\nonumber\\
+\eeq
+Note that this approximation, where the ensemble density matrix is
+optimized from a non-local exchange potential [rather than a local one,
+as expected from Eq.~(\ref{eq:var_ener_gokdft})] is applicable to real
+(three-dimension) systems. As readily seen from
+Eq.~(\ref{eq:eHF-dens_mat_func}), {\it ghost-interaction} errors will be
+introduced in the ensemble HF interaction energy:
+
+  
+
+In order to remove ghost interactions from the variational energy
+expression used in the first step, we then employ the (in-principle-exact)
+expression in Eq.~(\ref{eq:exact_ind_ener_OEP-like}). In this second
+step, the response of the individual density matrices to weight
+variations (last term on the right-hand side of
+Eq.~(\ref{eq:exact_ind_ener_OEP-like})) is neglected. The complete GIC
+procedure can be summarized as follows,    
+and
+
 In order to compute (approximate) energy levels within generalized
 GOK-DFT we use a two-step procedure. The first step consists in
 optimizing variationally the ensemble density matrix according to
@@ -438,27 +478,8 @@ c}[n]=\int d\br\;n(\br)\epsilon_{c}^{\bw}(n(\br)).
 \eeq
 More
 details about the construction of such a functional will be given in the
-following. In order to remove ghost interactions from the variational energy
-expression used in the first step, we then employ the (in-principle-exact)
-expression in Eq.~(\ref{eq:exact_ind_ener_OEP-like}). In this second
-step, the response of the individual density matrices to weight
-variations (last term on the right-hand side of
-Eq.~(\ref{eq:exact_ind_ener_OEP-like})) is neglected. The complete GIC
-procedure can be summarized as follows,    
-\beq
-{\bmg}^{\bw}\approx\argmin_{{\bm\gamma}^{\bw}}
-\Big\{
-{\rm
-Tr}\left[{\bm \gamma}^{{\bw}}{\bm h}\right]+W_{\rm
-HF}\left[{\bm\gamma}^{\bw}\right]
-+
-{E}^{{\bw}}_{\rm
-c}\left[n_{\bm\gamma^{\bw}}\right]
-%+E^{{\bw}}_{\rm c}\left[n_{\hat{\Gamma}^{{\bw}}}\right]
-\Big\},
-\nonumber\\
-\eeq
-and
+following. 
+
 \beq
 E^{(I)}&&\approx{\rm
 Tr}\left[{\bmg}^{(I)}{\bm h}\right]